and each morphismX(n+1)→X(n)X^{(n+1)} \to X^{(n)} induces an isomorphism on all homotopy groups in degree k≥(n+1)k \geq (n+1) (and the inclusion 1→πn(X(n))1 \to \pi_n(X^{(n)}) in degree nn as well as the identity 1=11 = 1 in degree k<nk \lt n).

The notion of Whitehead tower is dual to the notion of Postnikov tower, which instead is a factorization of the terminal morphism X→*X \to * into a tower, where homotopy groups are added from right to left.

In fact, the Whitehead tower may be constructed by taking each stage X(n+1)→X(n)X^{(n+1)} \to X^{(n)} to be the homotopy fiber of the corresponding map into the (n+1)(n+1)st stage of the Postnikov tower.

where each X⟨n⟩→X⟨n−1⟩X\langle n\rangle \to X\langle n-1 \rangle induces isomorphisms on homotopy groups πi\pi_i for i>ni\gt n and such that X⟨n⟩X\langle n\rangle is nn-connected (has trivial homotopy groups πi\pi_i for i≤ni \leq n). The homotopy long exact sequence then shows that the fiber of X⟨n⟩→X⟨n−1⟩X\langle n\rangle \to X\langle n-1 \rangle is a K(πn−1(X,x),n−1)K(\pi_{n-1}(X,x),n-1)Eilenberg-Mac Lane space. One has a model for K(πn−1(X,x),n−1)K(\pi_{n-1}(X,x),n-1) which is an abelian topological group; this has a remarkable consequence when (X,x)=(G,e)(X,x)=(G,e) is a topological group. Indeed, in this case one sees inductively that G⟨n⟩G\langle n\rangle has a model which is a topological group, which is an abelian group extension:

For instance, the string group can be realized as a topological group as a K(ℤ,2)K(\mathbb{Z},2)-extension of the spin group.

For n=0n=0 we require that X⟨0⟩↪XX\langle 0 \rangle \hookrightarrow X is the inclusion of the path-component of xx. Really this is defined up to homotopy, but we have a canonical model. If XX is locally connected and semilocally path-connected, then X⟨1⟩X\langle 1\rangle can be chosen as the universal covering space.

In traditional models this construction is highly non-functorial, except for nice spaces in low dimensions as remarked above.

Constructions

Whitehead’s construction

Whitehead 1952 answered the question, posed by Witold Hurewicz, of the existence of what we would now call nn-connected 'covers' of a given space XX, taking this to mean a fibration X⟨n⟩→XX\langle n\rangle \to X with X⟨n⟩X\langle n\ranglenn-connected and otherwise inducing isomorphisms on homotopy groups.

The construction proceeds as follows (using modern terminology). Given a pointed space (X,x)(X,x),

Choose a representative for the Postnikov sectionXnX_n such that X↪XnX \hookrightarrow X_n is a closed subspace (I would be tempted to make it a closed cofibration, but I don’t know any reason for this to be necessary -DMR).

Pull this back to XX, to get p:X⟨n⟩→Xp\colon X\langle n\rangle \to X, which is still a fibration. The induced maps on long exact sequences in homotopy can be compared, and show that pp has the desired properties.

This gives us a single nn-connected cover, but by considering the Postnikov tower

of XX, where each map X→XnX \to X_n is the inclusion of a closed subspace, it is simple to see there are induced maps X⟨n⟩→X⟨n−1⟩X\langle n\rangle \to X\langle n-1\rangle over XX for all nn.

One way of obtaining a Postnikov section as above is to choose representatives ϕg:Sn+1→X\phi_g\colon S^{n+1} \to X of generators gg of πn+1(X,x)\pi_{n+1}(X,x) and attaching cells: X(1)≔Bn+2∪{ϕg}XX(1)\coloneqq B^{n+2} \cup_{\{\phi_g\}} X. We then choose representatives for the generators of πn+2(X(1),x)\pi_{n+2}(X(1),x) and attach cells and so on. The colimit lim→nX(n)\lim_{\to n} X(n) is then a Postnikov section with the properties we require.

Understandably, this process is unbelievably non-canonical, and so we are generally reduced to existence theorems using this method – unless there is a functorial way to construct Postnikov sections. Strictly speaking we can only say annn-connected cover (except in special cases, like when n=1n=1 and XX is a well-connected space).

Functorial constructions

The nnth stage of the Whitehead tower of XX is the homotopy fiber of the map from XX to the nnth (or so) stage of its Postnikov tower, so one can use a functorial construction of the Postnikov tower plus a functorial construction of the homotopy fiber (such as the usual one using the path space of the target).

The nnth stage of the Whitehead tower of XX is also the cofibrant replacement for XX in the right Bousfield localization of Top with respect to the object SnS^n (or so). Since Top is right proper and cellular this localization exists by the result of chapter 5 of Hirschhorn’s book on localizations of model categories.

For instance w2w_2 can be identified as such by representing BO→τ≤2BO≃BO/∼nB O \to \tau_{\leq 2} B O \simeq BO/_{\sim_n} by a Kan fibration (see at Postnikov tower) between Kan complexes so that then the homotopy pullback (as discussed there) is given by an ordinary pullback. Since sSetsSet is a simplicial model category, sSet(S2,−)sSet(S^2,-) can be applied and preserves the pullback as well as the homotopy pullback, hence sends BO→τ≤2BO B O \to \tau_{\leq 2} B O to an isomorphism on connected components. This identifies BSO→B2ℤB SO \to B^2 \mathbb{Z} as being an isomorphism on the second homotopy group. Therefore, by the Hurewicz theorem, it is also an isomorphism on the cohomology groupH2(−,ℤ2)H^2(-,\mathbb{Z}_2). Analogously for the other characteristic maps.